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Acoustic 3D Plane Wave

The 1D plane wave can be extended into a 3D plane wave by considering a rectangular volume with coordinates x, y, z.

Continuity Equation, 3D

For a control volume, from the principle of conservation of mass, the instantaneous rate of change of mass in a control volume equals to the net mass flux flow into or out of the control volume, therefore:

As in 1D plane wave, the linearized form is :

 

And the time derivative of the equation is:

Euler's Equation of Inviscid Motion, 3D

For a control volume, from principle of momentum conservation, the instantaneous rate of change of net momentum of a control volume equals to the net applied force and the net momentum change due to the momentum flux flow into or out of the control volume. The applied force in this case is pressure only and no other forces, no gravity, no viscous force etc., then:

In x direction:

As in 1D plane wave, the linearized form is :

 

And the position derivative of the equation is

Similarly in y and z direction is

and

And get::

Equate two equations::

Wave Equation, 3D

Applying the equation of state and substitute the speed of propagation, imply.

Plane Wave in 3D

Assume harmonic sound wave, imply

substitute the harmonic function into the 3D wave equation and take the time differential operation, imply:

and get the Helmholtz equation:

Assume the pressure is of format:

substitute the pressure function into the Helmholtz equation, and take the Laplacian differential operation:

let the dispersion equation :

substitute into the Helmholtz equation:

By separating the variable, imply:

The corresponding solutions are:

substute all into the pressure function and get the solution of wave equation is:

Therefore a propagating wave can be decomposed into three components along three axes. A positive sign implies a forward wave from the source to the free space. While a negative sign implies a reflected wave. With the dispersion of the value, k into three components along the three axes